2D Coordinate Geometry

2D Coordinate Geometry: Level 3 Challenges


Circle OO is centered at the point of origin with point P=(3,4)P = (3 , 4) lying on it. The red line l:3x+4y7=0l : 3x + 4y - 7 = 0 intersects the circle at points AA and B,B, as shown.

What is the area of quadrilateral AOBP?AOBP?

{An+1=αAn+(1α)BnBn+1=αBn+(1α)CnCn+1=αCn+(1α)An \large {\left\{\begin{matrix}A_{n+1}=\alpha A_n+(1-\alpha)B_n \\ B_{n+1}=\alpha B_n+(1-\alpha)C_n \\ C_{n+1}=\alpha C_n+(1-\alpha)A_n\end{matrix}\right.}

Let A1,B1,C1A_1,B_1,C_1 be three distinct non-collinear points on the coordinate plane.
Also An,Bn,CnA_n,B_n,C_n satisfy the recurrence relation above (0<α<10<\alpha<1).

Then limnAn\displaystyle\lim_{n\to\infty}A_n is the __________\text{\_\_\_\_\_\_\_\_\_\_} of the triangle A1B1C1A_1B_1C_1.

Did you know that we can have coordinate systems where the coordinate axes are not perpendicular to each other? Such coordinate systems are known as oblique coordinate systems.

The above figure shows an oblique coordinate system. In such systems, the xx-coordinate of a point is found by measuring the distance from the point to the yy-axis parallel to the x x -axis. Similarly, the yy-coordinate of a point is found by measuring the distance from the point to the xx -axis parallel to the yy-axis. The red point in the above figure has the coordinates (a,b)(a, b).

Question: We are given two points: A (1,6)A \ (1,6) and B (5,2)B \ (5,2) in an oblique coordinate system where the angle between the positive axes is 60 60^{\circ} . What is the distance between the two points, ABAB?

The AOB\triangle AOB consists of point A=(0,1)A = (0 , 1), point O=(0,0)O = (0 , 0), and point BB lying somewhere on the xx-axis.

Let PP be the point in the first quadrant such that AP=PBAP = PB and AOPBAO \parallel PB, as shown above.

If the length of OPOP is 194\sqrt{194}, what is the area of the quadrilateral AOBP? AOBP?

A particle is on the point (3,5)(3,5) on the plane. It goes 10 units in the direction of positive xx-axis. Then it turns towards right making an angle of 90 degrees with its initial direction and goes ahead 5 units, Now it again takes a right turn and goes ahead 2.5 units. It keeps turning right and travels half the distance traveled in the turn just before it . This goes on without end. Find the final distance of the particle from the origin (0,0) (0,0) ?

The answer would come in the form square root of xx, submit your answer as xx.


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